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An upper-bound approach for equal channel angular extrusion with circular cross-section
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
journal homepage: www.elsevier.com/locate/jmatprotec
An upper-bound approach for equal channel angular extrusion with circular cross-section M.H. Paydar a,∗ , M. Reihanian a , R. Ebrahimi a , T.A. Dean b , M.M. Moshksar a a
Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran Department of Mechanical Engineering, School of Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
b
a r t i c l e
i n f o
a b s t r a c t
Article history:
In the present work, for the first time, an upper-bound approach is used to analyze the
Received 16 April 2006
equal channel angular extrusion with circular cross-sections. Based on this model the power
Received in revised form
dissipated on all frictional and velocity discontinuity surfaces are determined and the total
24 December 2006
power optimized analytically. To compare the theoretical results with experiments, an ECAE
Accepted 20 June 2007
die with a circular cross-section, having sharp corner was used to determine experimental extrusion force. The developed model predicts that the size of the plastic deformation zone and the relative extrusion pressure increase with increasing the constant friction factor.
Keywords:
In addition the results show that there is a good agreement between the theoretical and
Severe plastic deformation
experimental load displacement curve.
Equal channel angular extrusion
© 2007 Elsevier B.V. All rights reserved.
Upper-bound theory
1.
Introduction
Recently it has been shown that severe plastic deformation (SPD) is an effective method for producing nanostructured materials with full density (Valiev et al., 2000). Among all SPD methods, equal channel angular extrusion (ECAE), first presented by Segal et al. (1981), has received much attention because of the potential for producing large samples (Horita et al., 2001). During metal forming operations, prediction of load and the size of plastic deformation zone is an important task, because it can help in the design of tools. In the last decade, plastic deformation of materials during ECAE has been studied by many researchers. These studies are widely based on finite element methods (FEM) (Prangnell et al., 1997; Semiatin et al., 2000; Kim et al., 2001; Li et al., 2004; Bowen et al., 2000). Using these methods, the effect of die geometry, material properties and friction can be estimated. In addition to finite element methods, there are only a few methods
∗
Corresponding author. Tel.: +98 711 2307293; fax: +98 711 6287294. E-mail address: [email protected] (M.H. Paydar). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.06.051
that analytically analyze the ECAE process. These methods are generally based on the slip line theory (Segal, 2003) and upperbound analysis. Until now the upper-bound theory has been only used to analyze ECAE dies with square cross-sections (Lee, 2000; Alkotra and Sevillano, 2003; Altan et al., 2005; Eivani and Karimi Taheri, 2007). It is probably due to the complexity of dealing with circular cross-section ECAE dies. In the present work, an analytical model based on the upper-bound approach was developed to analyze ECAE process using a circular crosssection die. The power dissipated on all frictional and velocity discontinuity surfaces was determined and the total power optimized analytically. The effect of constant friction factor on the size of plastic deformation zone and relative pressure was investigated. To compare the theoretical results with the experimental ones, an ECAE die with circular cross-section was designed and a load–displacement curve was obtained using commercially pure aluminum as the raw material at room temperature and 200 ◦ C.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
2.
49
Analysis of deformation
Upper-bound is an analytical model for predicting the load that is equal to or higher than the actual force in metal forming operations. Based on this model, for a rigid–plastic Von-Misses material, among all kinematically admissible velocity fields, the actual one minimizes the power expressed by the equation below:
J = 2k V
1 ε˙ ε˙ dV + 2 ij ij
k ds +
Sv
mk ds −
Sf
Ti vi ds St
(1)
where k is the shear yield stress, ε˙ ij the components of strain rate, m the constant friction factor, V the volume of plastic deformation zone, Sv and Sf the area of velocity discontinuity and frictional surfaces, St the area where the tractions may occurs, the amount of velocity discontinuity on the frictional and discontinuity surfaces, vi and Ti are the velocity and tractions applied on the St , respectively. The first term in the equation above represents the power dissipated in the deformation zone. The second and third term express the power dissipated along the velocity discontinuity and frictional surfaces. The last term reflects the power due to the predetermined body tractions. The most important matter in understanding the model is to realize the geometry of the deformation zone. Fig. 1 shows an ECAE die with orthogonal circular cross-sections and a sharp outer arc of curvature. In this model, the volume considered for upper-bound analysis is bounded by OABCDEO, which is divided into four regions. In regions I and III, the materials move rigidly with the velocity of V0 . Region II, where the material is deformed plastically, is separated from regions I and III by the entrance and exit elliptical velocity discontinuity surfaces S1 and S2 (the ellipse OGBG O and OFDF O in Fig. 2), respectively. The minor and major diameters of the elliptical surfaces S1 and S2 are 2a (GG and FF in Fig. 2) and 2a/cos
Fig. 2 – Three-dimensional deformation model used in the upper-bound analysis of the ECAE process.
(OB and OD in Fig. 2), respectively, where a is the radius of the circular channel. The angle between the horizontal surface OA (or vertical surface OE) and surface S1 (or S2 ) is denoted by . Then the corner angle of plastic deformation zone ( ) is related to this angle by the equation: =
2
− 2
(2)
Based on this model, it is assumed that the plastic deformation zone (region II) is obtained by rotating the entrance elliptical surface S1 around point O from to (/2 − ). It is also assumed that in this region, the material moves along the concentric circles centered at point O. According to Fig. 2, it is obvious that the dead metal zone (region IV) is the region bounded between the space surface G BGFDF and the die walls. In region II the material moves with a constant velocity of V0 cos . Using a cylindrical coordinates, the assumed velocity field in region II is expressed as r = 0,
= V0 cos ,
Z = 0
(3)
Based on this velocity field, the only components of strain rate in the deformation zone are (Avitzur, 1968) ε˙ r = ε˙ r = −
1 V0 cos 2 r
(4)
where r is the radial distance from point O. The total power dissipated during ECAE process is obtained by ˙ total = W ˙ ideal + W ˙ S +W ˙ S +W ˙S ˙S ˙S W +W +W 1 2 AOB DOE dead zone ˙S ˙S +W +W channel wall GFOF G Fig. 1 – Two-dimensional deformation model used in the upper-bound analysis of the ECAE process.
(5)
˙ ideal is the power dissipated in the deformation where W ˙ S the power dissipated on the entrance and ˙ zone, WS1 and W 2
50
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
exit velocity discontinuity surfaces S1 and S2 , respectively. ˙S ˙S ˙S ,W and W are the powers dissipated on the W AOB DOE GFOF G cylindrical die walls AOB, DOE and GFOF G , respectively and ˙S is the power dissipated on the lateral surface of the W channel wall entrance channel due to friction. The detailed expression of the power dissipated on the all frictional and velocity discontinuity surfaces in Eq. (5) are presented in Appendix A. Using the equations of Appendix A, the total power dissipated during ECAE process is summarized as
˙ total = a2 kV0 W
+
2
− 2 2
− 2
2
2
+m
+ 2m
2
+ 4m tan + 2 tan
3 tan2
− 2
2
tan2
+1
2
Fig. 3 – Stress–strain curves of commercially pure aluminum at two temperatures.
cos
+1
cos
l
(6)
a
where m is the constant friction factor and l is the instant length of the billet in the entrance channel. The optimum value of angle that minimizes the total power can be obtained as below: ˙ total ∂W =0 ∂
(7)
which yields an expression between m and as m=
2 cos2 − 2 + 6 sin2 cos + 4 cos3 + (/2 − 2)(3 sin3 + 2 sin cos2 − 6 sin ) 4 − 2 sin2 cos − 4 cos3 + (/2 − 2)(2 sin − sin3 − 2 sin cos2 )
By equating the above power to the external power, the relative extrusion pressure for ECAE process with circular cross-section is obtained as P 1 = √
0 3 +
− 2 2
2
+m
2
√
− 2
2
where 0 =
3.
used as lubricants at 200 ◦ C and room temperature, respectively. The constant friction factor m, appropriate to the forming process, was estimated by the “Barrel Compression Test” (Ebrahimi and Najafizadeh, 2004). Tests were carried out at the same temperatures and using the same lubricants as for ECAE. A constant friction factor of 0.12 and 0.17 was measured at room temperature and 200 ◦ C, respectively. Compression tests were used to determine mean flow stress. Based on the stress–strain curves obtained from the compression test (Fig. 3), mean flow stresses at room temperature and 200 ◦ C were estimated as 160 MPa and 100 MPa, respectively.
2
+ 4m tan + 2 tan
3 tan2
−2
2
tan2 2
+1
cos
+1
cos +2m
l
a
(9)
4.
Results and discussion
4.1.
Effect of constant friction factor
(8)
Regarding Eq. (8), the optimized angle depends only on the constant friction factor m. Fig. 4 shows the variation of the optimized angle with constant friction factor m. It is seen that the optimized angle ϕ decreases with increasing the constant friction factor, m, which is due to the increase in size of the dead zone. In fact with increasing the constant friction
3k is the yield stress of the material.
Experimental procedure
Load–displacement curve was obtained experimentally to evaluate the theoretical results. To determine the experimental load–displacement curve, a sharp intersecting ECAE die set with an intersecting channel angle of 90◦ was designed. A commercial pure aluminum, annealed at 350 ◦ C for 3 h, was used as the raw material in this work. For ECAE processing, samples with 10 mm diameter and 30 mm length were machined from a rod. The ECAE process was performed at room temperature and 200 ◦ C to obtain two load–displacement curves. High temperature grease (graphite based) and oil were
Fig. 4 – Effect of constant friction factor on the optimized angle .
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
Fig. 5 – Effect of constant friction factor on the relative extrusion pressure.
Fig. 7 – Load–displacement curves at 200 ◦ C.
factor, the dead zone adopted a shape to minimize the contact frictional surfaces. Since the angle is limited between 0◦ and 45◦ , the reasonable values of the constant friction factor must be in the range of 0.03–0.5. This means that for constant friction factors less than 0.03, the deformation region (zone II) does not develop and the plastic deformation zone would be a plane along the intersection of the two channels ( = 45◦ ), which causes the strain extremely localized in a narrow deformation zone. But when the constant friction factor is more than 0.5, the optimized angle approaches to zero, and the plastic deformation zone develops to its largest size. Fig. 5 illustrates the variation of relative extrusion pressure with the constant friction factor m. As it is seen, friction has a strong effect on the relative extrusion pressure.
4.2. Comparison of theoretical and experimental load–displacement curve To determine the theoretical extrusion forces, the following procedure was applied. The external power applied for deformation of the material in ECAE process is given by ˙ ext = FV0 W
51
(10)
where F is the theoretical extrusion force and V0 is the Ram speed. By equating the external power and the total power obtained from Eq. (5), the theoretical extrusion force for ECAE process was obtained.
The theoretical results were compared with the experimental load–displacement curves in Figs. 6 and 7, at room temperature and 200 ◦ C, respectively. These results show a good agreement between the theoretical model and experiments. The gradual decrease in the experimental load–displacement curves is because of decreasing the frictional surface area in the entrance channel as the punch is advanced, but this fact was ignored in the present analysis.
5.
Conclusions
A new upper-bound model was used to analyze the ECAE process with circular cross-section channel, and the following results were obtained: 1. Based on the model, for constant friction factors less than 0.03, the deformation zone occurs in a plane bounded by the intersection of two channels. While for constant friction factors more than 0.5, the deformation zone develops to its largest size. 2. Friction has a strong effect on the relative extrusion pressure in ECAE, so that the relative extrusion pressure increases more than three times, when constant friction factor changes from 0.03 to 0.5. 3. The theoretical extrusion force obtained from the upperbound model was in a good agreement with the experimental results.
Acknowledgements The authors would like to acknowledge the financial support from the “Iranian Nanotechnology Initiative” and Shiraz University through the grant number of 85-GR-ENG-16.
Appendix A A.1. Power dissipated in the deformation zone Due to the given velocity field in Eq. (3), the only non-zero strain rate component in the cylindrical coordinate is
Fig. 6 – Load–displacement curves at room temperature.
ε˙ r = ε˙ r = −
V0 cos 2r
(A.1)
52
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
Based on the upper-bound theory, the power dissipated in the deformation zone (internal power) is given by
˙ ideal = 2k W V
1 ε˙ ε˙ dV 2 ij ij
(A.2)
where ε˙ ij ε˙ ij = 2˙ε2r
(A.3)
and dV is a volume element considered in the deformation zone (Fig. A1). Referring to Fig. A1, the volume element can be estimated as follows: dV = 2zr dr d
(A.4)
By considering the equation of an ellipse with the major and minor diameter of 2a and (2a/cos ), respectively (Fig. A1), the relation between z and r can be obtained as 2
2
z = (2ar cos − r cos )
1/2
(A.5)
Then the internal power becomes:
/2−
2a/cos
˙ ideal = 2k W
2
− 2
1/2
k ds = a2 kV0 tan
˙S =W ˙S = W 1 2
(A.7)
Sv
A.3. Power dissipated on the interface between the dead zone and the deformation zone The magnitude of velocity discontinuity on the surface between the dead zone and deformation zone is V0 cos . This surface is established in Fig. 2 as G BGFDF . The power dissipated on this surface is estimated as below:
˙S W = dead zone
r
0
The area of velocity discontinuity surfaces S1 and S2 is equal to the area of an ellipse with the minor and major diameters of 2a and (2a/cos ), respectively. The magnitude of the velocity discontinuity on these surfaces is V0 sin . Then:
k ds
(A.8)
SG BGFDF
where ds is a surface element on the interface between the dead zone and the deformation zone and is given by
V cos 0
× (2ar cos − r2 cos2 ) = a2 kV0
A.2. Power dissipated on the entrance and exit surfaces S1 and S2
ds =
r dr d (A.6)
2
− 2 (r sin2 + a cos )dr
(A.9)
then:
˙S =2 W G BGFDF
2a/cos
kV0 cos a/cos 2
= 2a kV0
2
− 2
2
− 2 (r sin2 + a cos )dr
3 tan2 +1 2
cos
(A.10)
A.4. Power dissipated on the die walls Based on the upper-bound approach, the power dissipated on a frictional surface is given by
˙f= W
mk ds
(A.11)
Sf
For cylindrical die walls AOB and DOE, = V0 s = 2a2 tan . Thus: ˙S ˙S W =W = 2a2 mkV0 tan AOB DOE
and
(A.12)
For die wall GFOF G : = V0 cos
(A.13)
and Fig. A1 – Volume element considered in the deformation zone. (a) Top view and (b) side view.
ds =
2
− 2 (r sin2 + a cos ) dr
(A.14)
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
then
a/cos
˙S = W GFOF G
mkV0 cos 0
= 2a2 mkV0
2
− 2
2
− 2 (r sin2 + a cos ) dr
tan2 2
+1
cos
(A.15)
For lateral surface of the entrance channel, = V0 and s = 2al. Thus: ˙S W = 2almkV0 channel wall
(A.16)
references
Alkotra, J., Sevillano, J.G., 2003. A comparison of fem and upper-bound type analysis of equal channel angular extrusion. J. Mater. Process. Technol. 141, 313–318. Altan, B.S., Purcek, G., Miskioglu, I., 2005. An upper-bound analysis for equal-channel angular extrusion. J. Mater. Process. Technol. 168, 137–146. Avitzur, B., 1968. Metal Forming: Processes and Analysis, 1st ed. McGraw-Hill, New York. Bowen, J.R., Gholina, A., Roberts, S.M., Prangnell, P.B., 2000. Analysis of the billet deformation behavior in equal channel angular extrusion. Mater. Sci. Eng. A 287, 87–99. Ebrahimi, R., Najafizadeh, A., 2004. A new method for evaluation of friction in bulk metal forming. J. Mater. Process. Technol. 152, 136–143.
53
Eivani, A.R., Karimi Taheri, A., 2007. An upper bound solution of ECAE process with outer curved corner. J. Mater. Process. Technol. 182, 555–563. Horita, Z., Fujinami, T., Langdon, T.G., 2001. The potential for scaling ECAP: effect of sample size on grain refinement and mechanical properties. Mater. Sci. Eng. A 318, 34–41. Kim, H.S., Seo, M.H., Hong, S.I., 2001. Plastic deformation analysis of metals during equal channel angular extrusion. J. Mater. Process. Technol. 113, 622–626. Lee, D.N., 2000. An upper-bound solution of channel angular deformation. Scripta Mater. 43, 115–118. Li, S., Bourke, M.A.M., Beyerlein, I.J., Alexander, D.J., Clausen, B., 2004. Finite element analysis of deformation zone and working load in equal channel angular extrusion. Mater. Sci. Eng. A 382, 217–236. Prangnell, P.B., Harris, C., Roberts, S.M., 1997. Finite element modeling of equal channel angular extrusion. Scripta Mater. 37, 983–989. Segal, V.M., 2003. Slip line solutions, deformation mode and loading history during equal channel angular extrusion. Mater. Sci. Eng. A 345, 36–46. Segal, V.M., Reznikov, V.I., Drobyshevski, A.E., Kopylov, V.I., 1981. Plastic working of metals by simple shear. Russ. Metall. 1, 99–105. Semiatin, S.L., Delo, D.P., Shell, E.B., 2000. The effect of material properties and tooling design on deformation and fracture during equal channel angular extrusion. Acta Mater. 48, 1841–1851. Valiev, R.Z., Islamgaliev, R.K., Alexandrov, I.V., 2000. Bulk nanostructured materials from severe plastic deformation. Prog. Mater. Sci. 45, 103–189.
journal homepage: www.elsevier.com/locate/jmatprotec
An upper-bound approach for equal channel angular extrusion with circular cross-section M.H. Paydar a,∗ , M. Reihanian a , R. Ebrahimi a , T.A. Dean b , M.M. Moshksar a a
Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran Department of Mechanical Engineering, School of Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
b
a r t i c l e
i n f o
a b s t r a c t
Article history:
In the present work, for the first time, an upper-bound approach is used to analyze the
Received 16 April 2006
equal channel angular extrusion with circular cross-sections. Based on this model the power
Received in revised form
dissipated on all frictional and velocity discontinuity surfaces are determined and the total
24 December 2006
power optimized analytically. To compare the theoretical results with experiments, an ECAE
Accepted 20 June 2007
die with a circular cross-section, having sharp corner was used to determine experimental extrusion force. The developed model predicts that the size of the plastic deformation zone and the relative extrusion pressure increase with increasing the constant friction factor.
Keywords:
In addition the results show that there is a good agreement between the theoretical and
Severe plastic deformation
experimental load displacement curve.
Equal channel angular extrusion
© 2007 Elsevier B.V. All rights reserved.
Upper-bound theory
1.
Introduction
Recently it has been shown that severe plastic deformation (SPD) is an effective method for producing nanostructured materials with full density (Valiev et al., 2000). Among all SPD methods, equal channel angular extrusion (ECAE), first presented by Segal et al. (1981), has received much attention because of the potential for producing large samples (Horita et al., 2001). During metal forming operations, prediction of load and the size of plastic deformation zone is an important task, because it can help in the design of tools. In the last decade, plastic deformation of materials during ECAE has been studied by many researchers. These studies are widely based on finite element methods (FEM) (Prangnell et al., 1997; Semiatin et al., 2000; Kim et al., 2001; Li et al., 2004; Bowen et al., 2000). Using these methods, the effect of die geometry, material properties and friction can be estimated. In addition to finite element methods, there are only a few methods
∗
Corresponding author. Tel.: +98 711 2307293; fax: +98 711 6287294. E-mail address: [email protected] (M.H. Paydar). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.06.051
that analytically analyze the ECAE process. These methods are generally based on the slip line theory (Segal, 2003) and upperbound analysis. Until now the upper-bound theory has been only used to analyze ECAE dies with square cross-sections (Lee, 2000; Alkotra and Sevillano, 2003; Altan et al., 2005; Eivani and Karimi Taheri, 2007). It is probably due to the complexity of dealing with circular cross-section ECAE dies. In the present work, an analytical model based on the upper-bound approach was developed to analyze ECAE process using a circular crosssection die. The power dissipated on all frictional and velocity discontinuity surfaces was determined and the total power optimized analytically. The effect of constant friction factor on the size of plastic deformation zone and relative pressure was investigated. To compare the theoretical results with the experimental ones, an ECAE die with circular cross-section was designed and a load–displacement curve was obtained using commercially pure aluminum as the raw material at room temperature and 200 ◦ C.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
2.
49
Analysis of deformation
Upper-bound is an analytical model for predicting the load that is equal to or higher than the actual force in metal forming operations. Based on this model, for a rigid–plastic Von-Misses material, among all kinematically admissible velocity fields, the actual one minimizes the power expressed by the equation below:
J = 2k V
1 ε˙ ε˙ dV + 2 ij ij
k ds +
Sv
mk ds −
Sf
Ti vi ds St
(1)
where k is the shear yield stress, ε˙ ij the components of strain rate, m the constant friction factor, V the volume of plastic deformation zone, Sv and Sf the area of velocity discontinuity and frictional surfaces, St the area where the tractions may occurs, the amount of velocity discontinuity on the frictional and discontinuity surfaces, vi and Ti are the velocity and tractions applied on the St , respectively. The first term in the equation above represents the power dissipated in the deformation zone. The second and third term express the power dissipated along the velocity discontinuity and frictional surfaces. The last term reflects the power due to the predetermined body tractions. The most important matter in understanding the model is to realize the geometry of the deformation zone. Fig. 1 shows an ECAE die with orthogonal circular cross-sections and a sharp outer arc of curvature. In this model, the volume considered for upper-bound analysis is bounded by OABCDEO, which is divided into four regions. In regions I and III, the materials move rigidly with the velocity of V0 . Region II, where the material is deformed plastically, is separated from regions I and III by the entrance and exit elliptical velocity discontinuity surfaces S1 and S2 (the ellipse OGBG O and OFDF O in Fig. 2), respectively. The minor and major diameters of the elliptical surfaces S1 and S2 are 2a (GG and FF in Fig. 2) and 2a/cos
Fig. 2 – Three-dimensional deformation model used in the upper-bound analysis of the ECAE process.
(OB and OD in Fig. 2), respectively, where a is the radius of the circular channel. The angle between the horizontal surface OA (or vertical surface OE) and surface S1 (or S2 ) is denoted by . Then the corner angle of plastic deformation zone ( ) is related to this angle by the equation: =
2
− 2
(2)
Based on this model, it is assumed that the plastic deformation zone (region II) is obtained by rotating the entrance elliptical surface S1 around point O from to (/2 − ). It is also assumed that in this region, the material moves along the concentric circles centered at point O. According to Fig. 2, it is obvious that the dead metal zone (region IV) is the region bounded between the space surface G BGFDF and the die walls. In region II the material moves with a constant velocity of V0 cos . Using a cylindrical coordinates, the assumed velocity field in region II is expressed as r = 0,
= V0 cos ,
Z = 0
(3)
Based on this velocity field, the only components of strain rate in the deformation zone are (Avitzur, 1968) ε˙ r = ε˙ r = −
1 V0 cos 2 r
(4)
where r is the radial distance from point O. The total power dissipated during ECAE process is obtained by ˙ total = W ˙ ideal + W ˙ S +W ˙ S +W ˙S ˙S ˙S W +W +W 1 2 AOB DOE dead zone ˙S ˙S +W +W channel wall GFOF G Fig. 1 – Two-dimensional deformation model used in the upper-bound analysis of the ECAE process.
(5)
˙ ideal is the power dissipated in the deformation where W ˙ S the power dissipated on the entrance and ˙ zone, WS1 and W 2
50
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
exit velocity discontinuity surfaces S1 and S2 , respectively. ˙S ˙S ˙S ,W and W are the powers dissipated on the W AOB DOE GFOF G cylindrical die walls AOB, DOE and GFOF G , respectively and ˙S is the power dissipated on the lateral surface of the W channel wall entrance channel due to friction. The detailed expression of the power dissipated on the all frictional and velocity discontinuity surfaces in Eq. (5) are presented in Appendix A. Using the equations of Appendix A, the total power dissipated during ECAE process is summarized as
˙ total = a2 kV0 W
+
2
− 2 2
− 2
2
2
+m
+ 2m
2
+ 4m tan + 2 tan
3 tan2
− 2
2
tan2
+1
2
Fig. 3 – Stress–strain curves of commercially pure aluminum at two temperatures.
cos
+1
cos
l
(6)
a
where m is the constant friction factor and l is the instant length of the billet in the entrance channel. The optimum value of angle that minimizes the total power can be obtained as below: ˙ total ∂W =0 ∂
(7)
which yields an expression between m and as m=
2 cos2 − 2 + 6 sin2 cos + 4 cos3 + (/2 − 2)(3 sin3 + 2 sin cos2 − 6 sin ) 4 − 2 sin2 cos − 4 cos3 + (/2 − 2)(2 sin − sin3 − 2 sin cos2 )
By equating the above power to the external power, the relative extrusion pressure for ECAE process with circular cross-section is obtained as P 1 = √
0 3 +
− 2 2
2
+m
2
√
− 2
2
where 0 =
3.
used as lubricants at 200 ◦ C and room temperature, respectively. The constant friction factor m, appropriate to the forming process, was estimated by the “Barrel Compression Test” (Ebrahimi and Najafizadeh, 2004). Tests were carried out at the same temperatures and using the same lubricants as for ECAE. A constant friction factor of 0.12 and 0.17 was measured at room temperature and 200 ◦ C, respectively. Compression tests were used to determine mean flow stress. Based on the stress–strain curves obtained from the compression test (Fig. 3), mean flow stresses at room temperature and 200 ◦ C were estimated as 160 MPa and 100 MPa, respectively.
2
+ 4m tan + 2 tan
3 tan2
−2
2
tan2 2
+1
cos
+1
cos +2m
l
a
(9)
4.
Results and discussion
4.1.
Effect of constant friction factor
(8)
Regarding Eq. (8), the optimized angle depends only on the constant friction factor m. Fig. 4 shows the variation of the optimized angle with constant friction factor m. It is seen that the optimized angle ϕ decreases with increasing the constant friction factor, m, which is due to the increase in size of the dead zone. In fact with increasing the constant friction
3k is the yield stress of the material.
Experimental procedure
Load–displacement curve was obtained experimentally to evaluate the theoretical results. To determine the experimental load–displacement curve, a sharp intersecting ECAE die set with an intersecting channel angle of 90◦ was designed. A commercial pure aluminum, annealed at 350 ◦ C for 3 h, was used as the raw material in this work. For ECAE processing, samples with 10 mm diameter and 30 mm length were machined from a rod. The ECAE process was performed at room temperature and 200 ◦ C to obtain two load–displacement curves. High temperature grease (graphite based) and oil were
Fig. 4 – Effect of constant friction factor on the optimized angle .
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
Fig. 5 – Effect of constant friction factor on the relative extrusion pressure.
Fig. 7 – Load–displacement curves at 200 ◦ C.
factor, the dead zone adopted a shape to minimize the contact frictional surfaces. Since the angle is limited between 0◦ and 45◦ , the reasonable values of the constant friction factor must be in the range of 0.03–0.5. This means that for constant friction factors less than 0.03, the deformation region (zone II) does not develop and the plastic deformation zone would be a plane along the intersection of the two channels ( = 45◦ ), which causes the strain extremely localized in a narrow deformation zone. But when the constant friction factor is more than 0.5, the optimized angle approaches to zero, and the plastic deformation zone develops to its largest size. Fig. 5 illustrates the variation of relative extrusion pressure with the constant friction factor m. As it is seen, friction has a strong effect on the relative extrusion pressure.
4.2. Comparison of theoretical and experimental load–displacement curve To determine the theoretical extrusion forces, the following procedure was applied. The external power applied for deformation of the material in ECAE process is given by ˙ ext = FV0 W
51
(10)
where F is the theoretical extrusion force and V0 is the Ram speed. By equating the external power and the total power obtained from Eq. (5), the theoretical extrusion force for ECAE process was obtained.
The theoretical results were compared with the experimental load–displacement curves in Figs. 6 and 7, at room temperature and 200 ◦ C, respectively. These results show a good agreement between the theoretical model and experiments. The gradual decrease in the experimental load–displacement curves is because of decreasing the frictional surface area in the entrance channel as the punch is advanced, but this fact was ignored in the present analysis.
5.
Conclusions
A new upper-bound model was used to analyze the ECAE process with circular cross-section channel, and the following results were obtained: 1. Based on the model, for constant friction factors less than 0.03, the deformation zone occurs in a plane bounded by the intersection of two channels. While for constant friction factors more than 0.5, the deformation zone develops to its largest size. 2. Friction has a strong effect on the relative extrusion pressure in ECAE, so that the relative extrusion pressure increases more than three times, when constant friction factor changes from 0.03 to 0.5. 3. The theoretical extrusion force obtained from the upperbound model was in a good agreement with the experimental results.
Acknowledgements The authors would like to acknowledge the financial support from the “Iranian Nanotechnology Initiative” and Shiraz University through the grant number of 85-GR-ENG-16.
Appendix A A.1. Power dissipated in the deformation zone Due to the given velocity field in Eq. (3), the only non-zero strain rate component in the cylindrical coordinate is
Fig. 6 – Load–displacement curves at room temperature.
ε˙ r = ε˙ r = −
V0 cos 2r
(A.1)
52
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
Based on the upper-bound theory, the power dissipated in the deformation zone (internal power) is given by
˙ ideal = 2k W V
1 ε˙ ε˙ dV 2 ij ij
(A.2)
where ε˙ ij ε˙ ij = 2˙ε2r
(A.3)
and dV is a volume element considered in the deformation zone (Fig. A1). Referring to Fig. A1, the volume element can be estimated as follows: dV = 2zr dr d
(A.4)
By considering the equation of an ellipse with the major and minor diameter of 2a and (2a/cos ), respectively (Fig. A1), the relation between z and r can be obtained as 2
2
z = (2ar cos − r cos )
1/2
(A.5)
Then the internal power becomes:
/2−
2a/cos
˙ ideal = 2k W
2
− 2
1/2
k ds = a2 kV0 tan
˙S =W ˙S = W 1 2
(A.7)
Sv
A.3. Power dissipated on the interface between the dead zone and the deformation zone The magnitude of velocity discontinuity on the surface between the dead zone and deformation zone is V0 cos . This surface is established in Fig. 2 as G BGFDF . The power dissipated on this surface is estimated as below:
˙S W = dead zone
r
0
The area of velocity discontinuity surfaces S1 and S2 is equal to the area of an ellipse with the minor and major diameters of 2a and (2a/cos ), respectively. The magnitude of the velocity discontinuity on these surfaces is V0 sin . Then:
k ds
(A.8)
SG BGFDF
where ds is a surface element on the interface between the dead zone and the deformation zone and is given by
V cos 0
× (2ar cos − r2 cos2 ) = a2 kV0
A.2. Power dissipated on the entrance and exit surfaces S1 and S2
ds =
r dr d (A.6)
2
− 2 (r sin2 + a cos )dr
(A.9)
then:
˙S =2 W G BGFDF
2a/cos
kV0 cos a/cos 2
= 2a kV0
2
− 2
2
− 2 (r sin2 + a cos )dr
3 tan2 +1 2
cos
(A.10)
A.4. Power dissipated on the die walls Based on the upper-bound approach, the power dissipated on a frictional surface is given by
˙f= W
mk ds
(A.11)
Sf
For cylindrical die walls AOB and DOE, = V0 s = 2a2 tan . Thus: ˙S ˙S W =W = 2a2 mkV0 tan AOB DOE
and
(A.12)
For die wall GFOF G : = V0 cos
(A.13)
and Fig. A1 – Volume element considered in the deformation zone. (a) Top view and (b) side view.
ds =
2
− 2 (r sin2 + a cos ) dr
(A.14)
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 48–53
then
a/cos
˙S = W GFOF G
mkV0 cos 0
= 2a2 mkV0
2
− 2
2
− 2 (r sin2 + a cos ) dr
tan2 2
+1
cos
(A.15)
For lateral surface of the entrance channel, = V0 and s = 2al. Thus: ˙S W = 2almkV0 channel wall
(A.16)
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